We study the bias of classical quantile regression and instrumental variable quantile regression estimators. While being asymptotically first-order unbiased, these estimators can have non-negligible second-order biases. We derive a higher-order stochastic expansion of these estimators using empirical process theory. Based on this expansion, we derive an explicit formula for the second-order bias and propose a feasible bias correction procedure that uses finite-difference estimators of the bias components. The proposed bias correction method performs well in simulations. We provide an empirical illustration using Engel's classical data on household expenditure.

翻译：我们研究古典的四分位回归和可变的四分位回归估计值的偏向性。这些估计值虽然在第一阶上没有偏向,但可能有不可忽略的第二阶偏向。我们利用经验过程理论对这些估计值进行更高层次的随机扩张。根据这种扩张,我们为第二阶偏向得出一个明确的公式,并提出可行的偏向纠正程序,使用对偏向部分的有限偏向估计值。提议的偏向纠正方法在模拟中表现良好。我们用Engel关于家庭支出的古典数据提供经验性说明。